[PPT] - LMI results for robust control design of observer-based controllers, PowerPoint Presentation

SLIDE 1

LMI results for robust control design of observer-based controllers, the discrete-time case with polytopic uncertainties

Dimitri PEAUCELLE Yoshio EBIHARA IFAC World Congress Wednesday August 25, 2014, Cape Town

SLIDE 2

Introduction ■ Curiosity at the origin of this work:

Many existing LMI results for robust state-feedback design
The Luenberger type observer problem is “dual" to state-feedback, but... few LMI results
Many results for robust output filtering issue, but applicable only to stable plants
D. Peaucelle

1 Cape Town, August 2014

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Introduction ■ Issues raised by the study of robust observers: xk+1 = A(θ)xk + Buk yk = Cxk , ˆ xk+1 = Aoˆ xk + Buk + L(Cˆ xk − yk) ek = xk − ˆ xk

What model Ao for the observer ?

▲ Usual answer is to decompose a priori A(θ) = Ao + ∆(θ) ▲ For example in NL observers, Ao models integrators in series

One cannot expect ek2>k1 = 0 even if ek1 = 0
Separation principle for the design of state-feedback and observer gains?

  xk+1 ek+1   =   A(θ) + BK −BK A(θ) − Ao Ao + LC     xk ek  

D. Peaucelle

2 Cape Town, August 2014

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Outline   xk+1 ek+1   =   A(θ) + BK −BK A(θ) − Ao Ao + LC     xk ek   ■ S-variable approach to state-feedback design, and analysis of the closed-loop ■ Joint model and gain observer design with minimization of influence on state-feedback ■ Robust analysis of the state-feedback + observer feedback loop ■ Numerical example ■ Conclusions

D. Peaucelle

3 Cape Town, August 2014

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state-feedback design ■ Discrete-time systems with polytopic uncertainties A(θ) =

¯ v

v=1

θvA[v] : θv=1...¯

v ≥ 0 , ¯ v

v=1

θv = 1 ■ Example of existing LMI result for state-feedback design (SFdesign)

∃P [v]

1

≻ 0 ∃F1 ∃ ˆ K

:   

P [v]

1

BwBT

w − P [v] 1

−µ2

∞I

   ≺        

F1 −(A[v]F1 + B ˆ K) −C[v]

z F1

  

I


   

S

.

If (SFdesign) hold for all vertices then K = ˆ

KF −1

1

robustly stabilizes the plant and

xk+1 = (A(θ) + BK)xk + Bwwk zk = Cz(θ)xk

has H∞ norm robustly less than µ∞, i.e. z2 ≤ µ∞w2.

D. Peaucelle

4 Cape Town, August 2014

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state-feedback analysis ■ Robust analysis LMI result for the plant with state-feedback (SFanalysis) ∃P [v]

3

≻ 0 ∃G3 ∃Q ≻ 0 :     P [v]

3

Q − P [v]

3

−I     ≺

G3
I

− (A[v] + BK) B S .

If K is solution to (SFdesign) then (SFanalysis) is feasible
If (SFanalysis) hold for all vertices then the following plant is robustly stable and

xk+1 = (A(θ) + BK)xk − BKek gk = Q1/2xk

has H∞ norm robustly less than 1, i.e. Q−1/2x2 ≤ Kek2.

Virtual output gk models the excursions of the state due to perturbations on the control.
Maximizing Q gives indications on the maximal excusions.
D. Peaucelle

5 Cape Town, August 2014

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Joint model and gain observer design ■ LMI result for robust observer design (Odesign) ∃P [v]

42 ≻ 0, ∃P [v] 4p KT K, ∃F4, ∃ ˆ

Ao, ∃ˆ L :   

P [v]

42

KT K − P [v]

42

−γ2

2Q

   ≺        

I

  

F4

− ( ˆ Ao + ˆ LC) ˆ Ao − F4A[v]



   

S

  

P [v]

4p

−P [v]

4p

−γ2

pQ

   ≺        

I

  

F4

− ( ˆ Ao + ˆ LC) ˆ Ao − F4A[v]



   

S

For all K and Q the LMI problem (Odesign) is feasible.
(Odesign) provides Ao = F −1

4

ˆ Ao, L = F −1

4

ˆ L s.t. the error dynamics are stable ek+1 = (Ao + LC)ek + (A(θ) − Ao)xk

and guarantees the following robust properties:

Ke2 ≤ γ2Q−1x2 , Kep = max

k

Kek ≤ γpQ−1x2

D. Peaucelle

6 Cape Town, August 2014

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Closed-loop analysis ■ Small gain theorem guarantees closed-loop robust stability if γ2 < 1.

Minimizing γ2 improves stability,

but tends to give high gain observers with large peak responses.

Minimizing a linear combination of γ2 and γp gives a trade-off between the two effects
D. Peaucelle

7 Cape Town, August 2014

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Closed-loop analysis ■ Robust analysis LMI result for the closed-loop plant (Oanalysis) ∃P [v]

6

≻ 0, ∃G6 :     

P [v]

6

  C[v]T

z

    C[v]

z

 

T

− P [v]

6

−ν∞2I

     ≺

G6
I 0

−A[v] −BK −Bw 0 I LC −Ao − BK − LC

S .

If (Oanalysis) hold for all vertices then the state-fedback + observer loop robustly stabilizes

the plant and

xk+1 = A(θ)xk + Buk + Bwwk, ˆ xk+1 = (Ao + BK + LC)ˆ xk − Lyk yk = Cxk, zk = Cz(θ)xk uk = Kˆ xk

has H∞ norm robustly less than ν∞, i.e. z2 ≤ ν∞ ˆ

w2.

One can expect µ∞ ≤ ν∞, i.e. that the observer-based control degrades the performance

compared to the ideal state-feedback.

D. Peaucelle

8 Cape Town, August 2014

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Numerical example xk+1 =   a b 1   xk +   1   uk +   0.1   wk, yk =

1
xk

zk =

1
xk

■ a ∈ [ 0.9, 1.1 ] and b ∈ [ 0.9, 1.1 ], i.e. ¯ v = 4 vertices. None of the vertices are stable.

(SFdesign) with µ∞ = 1 gives K =
−1.0633

−1.0324

.
(SFanalysis) with max Tr(Q) gives Q =

  0.1239 0.0527 0.0527 0.5730  .

(Odesign) with min γ2 + γp gives γ2 = 0.8797, γp = 0.8575 and

Ao =   0.9946 0.9807 0.9946 −0.0191   , L =   −2.3637 −1.3565   . ▲ γ2 < 1, the closed-loop is robustly stable ▲ Ao =   1 1 1  

(Oanalysis) with min ν∞ gives ν∞ = 1.0268.
D. Peaucelle

9 Cape Town, August 2014

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Numerical example

Impulse responses (for several random values of uncertainties)

1 2 3 4 5 6 7 8 9 10 −0.02 0.02 0.04 0.06 0.08 0.1 Impulse Response Time (seconds) Amplitude 5 10 15 20 25 30 35 −0.15 −0.1 −0.05 0.05 0.1 0.15 0.2 0.25 Impulse Response Time (seconds) Amplitude

ideal state-feedback with observer-based control

D. Peaucelle

10 Cape Town, August 2014

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Numerical example

Control inputs for two different choices of observers

5 10 15 20 25 30 35 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 Impulse Response Time (seconds) Amplitude 10 20 30 40 50 60 −0.3 −0.2 −0.1 0.1 Impulse Response Time (seconds) Amplitude

(Odesign) with min γ2 + γp (Odesign) with min γ2 + 104γp

Allows to reduce the peak response but with slower convergence
D. Peaucelle

11 Cape Town, August 2014

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Conclusions ■ Robust observer design revisited ■ Proposed heuristic based on LMIs only ■ Observer optimized not to perturb too strongly the given state-feedback

Tradeoff between L2 and peak criteria

■ Discussions about S-variable approach LMIs:

primal (observer case) VS dual (state-feedback) analysis (G S-variable) VS design (F structured S-variable) The S-Variable Approach to LMI-Based Robust Control Springer, Y. Ebihara, D. Peaucelle, D. Arzelier, 2015

D. Peaucelle